Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F

Percentage Accurate: 93.5% → 99.2%

Time: 12.3s

Alternatives: 12

Speedup: 1.1×

• 25.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x - \frac{y \cdot \left(z - t\right)}{a} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (- x (/ (* y (- z t)) a)))

C

double code(double x, double y, double z, double t, double a) {
	return x - ((y * (z - t)) / a);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x - ((y * (z - t)) / a)
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x - ((y * (z - t)) / a);
}

Python

def code(x, y, z, t, a):
	return x - ((y * (z - t)) / a)

Julia

function code(x, y, z, t, a)
	return Float64(x - Float64(Float64(y * Float64(z - t)) / a))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x - ((y * (z - t)) / a);
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x - N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]

Initial Program: 93.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x - \frac{y \cdot \left(z - t\right)}{a} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (- x (/ (* y (- z t)) a)))

C

double code(double x, double y, double z, double t, double a) {
	return x - ((y * (z - t)) / a);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x - ((y * (z - t)) / a)
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x - ((y * (z - t)) / a);
}

Python

def code(x, y, z, t, a):
	return x - ((y * (z - t)) / a)

Julia

function code(x, y, z, t, a)
	return Float64(x - Float64(Float64(y * Float64(z - t)) / a))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x - ((y * (z - t)) / a);
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x - N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+203}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+120}:\\ \;\;\;\;x - \frac{t\_1}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (- z t))))
   (if (<= t_1 -1e+203)
     (fma (/ (- t z) a) y x)
     (if (<= t_1 1e+120) (- x (/ t_1 a)) (fma (/ y a) (- t z) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * (z - t);
	double tmp;
	if (t_1 <= -1e+203) {
		tmp = fma(((t - z) / a), y, x);
	} else if (t_1 <= 1e+120) {
		tmp = x - (t_1 / a);
	} else {
		tmp = fma((y / a), (t - z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(z - t))
	tmp = 0.0
	if (t_1 <= -1e+203)
		tmp = fma(Float64(Float64(t - z) / a), y, x);
	elseif (t_1 <= 1e+120)
		tmp = Float64(x - Float64(t_1 / a));
	else
		tmp = fma(Float64(y / a), Float64(t - z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+203], N[(N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 1e+120], N[(x - N[(t$95$1 / a), $MachinePrecision]), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * N[(t - z), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y (-.f64 z t)) < -9.9999999999999999e202

   1. Initial program 81.0%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]

   3. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - z}{a}, y, x\right)} \]

   if -9.9999999999999999e202 < (*.f64 y (-.f64 z t)) < 9.9999999999999998e119

   1. Initial program 99.4%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]

   if 9.9999999999999998e119 < (*.f64 y (-.f64 z t))

   1. Initial program 88.2%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]

   3. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 97.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \frac{y \cdot \left(z - t\right)}{a} \leq 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (- x (/ (* y (- z t)) a)) 1e-19)
   (fma (/ (- t z) a) y x)
   (fma (/ y a) (- t z) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x - ((y * (z - t)) / a)) <= 1e-19) {
		tmp = fma(((t - z) / a), y, x);
	} else {
		tmp = fma((y / a), (t - z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x - Float64(Float64(y * Float64(z - t)) / a)) <= 1e-19)
		tmp = fma(Float64(Float64(t - z) / a), y, x);
	else
		tmp = fma(Float64(y / a), Float64(t - z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x - N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], 1e-19], N[(N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * N[(t - z), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (/.f64 (*.f64 y (-.f64 z t)) a)) < 9.9999999999999998e-20

   1. Initial program 94.0%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]

   3. Applied rewrites 94.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - z}{a}, y, x\right)} \]

   if 9.9999999999999998e-20 < (-.f64 x (/.f64 (*.f64 y (-.f64 z t)) a))

   1. Initial program 92.9%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]

   3. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 96.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{y}{a}, t - z, x\right) \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (fma (/ y a) (- t z) x))

C

double code(double x, double y, double z, double t, double a) {
	return fma((y / a), (t - z), x);
}

Julia

function code(x, y, z, t, a)
	return fma(Float64(y / a), Float64(t - z), x)
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(y / a), $MachinePrecision] * N[(t - z), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 93.5%

   \[x - \frac{y \cdot \left(z - t\right)}{a} \]

3. Applied rewrites 97.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)} \]
4. Add Preprocessing

Alternative 4: 84.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+114}:\\ \;\;\;\;\left(t - z\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;t\_1 \leq 10000:\\ \;\;\;\;x - \frac{y \cdot z}{a}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+183}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t - z}{a} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) a)))
   (if (<= t_1 -5e+114)
     (* (- t z) (/ y a))
     (if (<= t_1 10000.0)
       (- x (/ (* y z) a))
       (if (<= t_1 2e+183) (fma (/ y a) t x) (* (/ (- t z) a) y))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (z - t)) / a;
	double tmp;
	if (t_1 <= -5e+114) {
		tmp = (t - z) * (y / a);
	} else if (t_1 <= 10000.0) {
		tmp = x - ((y * z) / a);
	} else if (t_1 <= 2e+183) {
		tmp = fma((y / a), t, x);
	} else {
		tmp = ((t - z) / a) * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y * Float64(z - t)) / a)
	tmp = 0.0
	if (t_1 <= -5e+114)
		tmp = Float64(Float64(t - z) * Float64(y / a));
	elseif (t_1 <= 10000.0)
		tmp = Float64(x - Float64(Float64(y * z) / a));
	elseif (t_1 <= 2e+183)
		tmp = fma(Float64(y / a), t, x);
	else
		tmp = Float64(Float64(Float64(t - z) / a) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+114], N[(N[(t - z), $MachinePrecision] * N[(y / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 10000.0], N[(x - N[(N[(y * z), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+183], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], N[(N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] * y), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -5.0000000000000001e114

   1. Initial program 87.9%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot \left(z - t\right)}{a}} \]

   4. Applied rewrites 89.7%

      \[\leadsto \color{blue}{\left(t - z\right) \cdot \frac{y}{a}} \]

   if -5.0000000000000001e114 < (/.f64 (*.f64 y (-.f64 z t)) a) < 1e4

   1. Initial program 99.2%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]

   2. Taylor expanded in z around inf

      \[\leadsto x - \frac{y \cdot \color{blue}{z}}{a} \]
   3. Step-by-step derivation

   4. Applied rewrites 85.5%

      \[\leadsto x - \frac{y \cdot \color{blue}{z}}{a} \]

   if 1e4 < (/.f64 (*.f64 y (-.f64 z t)) a) < 1.99999999999999989e183

   1. Initial program 99.7%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]

   3. Applied rewrites 97.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)} \]

   4. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\frac{y}{a}, \color{blue}{t}, x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 63.7%

      \[\leadsto \mathsf{fma}\left(\frac{y}{a}, \color{blue}{t}, x\right) \]

   if 1.99999999999999989e183 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 85.1%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]

   3. Applied rewrites 97.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y \cdot \left(t - z\right)}{a}} \]
   5. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto y \cdot \color{blue}{\frac{t - z}{a}} \]

      2. sub-div N/A

         \[\leadsto y \cdot \left(\frac{t}{a} - \color{blue}{\frac{z}{a}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\frac{t}{a} - \frac{z}{a}\right) \cdot \color{blue}{y} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{t}{a} - \frac{z}{a}\right) \cdot \color{blue}{y} \]

      5. sub-div N/A

         \[\leadsto \frac{t - z}{a} \cdot y \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{t - z}{a} \cdot y \]

      7. lift--.f64 88.4

         \[\leadsto \frac{t - z}{a} \cdot y \]

   6. Applied rewrites 88.4%

      \[\leadsto \color{blue}{\frac{t - z}{a} \cdot y} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 84.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+98}:\\ \;\;\;\;\left(t - z\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+183}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t - z}{a} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) a)))
   (if (<= t_1 -1e+98)
     (* (- t z) (/ y a))
     (if (<= t_1 2e+183) (fma y (/ t a) x) (* (/ (- t z) a) y)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (z - t)) / a;
	double tmp;
	if (t_1 <= -1e+98) {
		tmp = (t - z) * (y / a);
	} else if (t_1 <= 2e+183) {
		tmp = fma(y, (t / a), x);
	} else {
		tmp = ((t - z) / a) * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y * Float64(z - t)) / a)
	tmp = 0.0
	if (t_1 <= -1e+98)
		tmp = Float64(Float64(t - z) * Float64(y / a));
	elseif (t_1 <= 2e+183)
		tmp = fma(y, Float64(t / a), x);
	else
		tmp = Float64(Float64(Float64(t - z) / a) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+98], N[(N[(t - z), $MachinePrecision] * N[(y / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+183], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -9.99999999999999998e97

   1. Initial program 88.3%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot \left(z - t\right)}{a}} \]

   4. Applied rewrites 88.9%

      \[\leadsto \color{blue}{\left(t - z\right) \cdot \frac{y}{a}} \]

   if -9.99999999999999998e97 < (/.f64 (*.f64 y (-.f64 z t)) a) < 1.99999999999999989e183

   1. Initial program 99.3%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x - -1 \cdot \frac{t \cdot y}{a}} \]
   3. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x - \left(\mathsf{neg}\left(\frac{t \cdot y}{a}\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto x - \left(\mathsf{neg}\left(t \cdot \frac{y}{a}\right)\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto x - \left(\mathsf{neg}\left(t\right)\right) \cdot \color{blue}{\frac{y}{a}} \]

      4. fp-cancel-sign-sub-inv N/A

         \[\leadsto x + \color{blue}{t \cdot \frac{y}{a}} \]

      5. associate-/l* N/A

         \[\leadsto x + \frac{t \cdot y}{\color{blue}{a}} \]

      6. +-commutative N/A

         \[\leadsto \frac{t \cdot y}{a} + \color{blue}{x} \]

      7. *-commutative N/A

         \[\leadsto \frac{y \cdot t}{a} + x \]

      8. associate-/l* N/A

         \[\leadsto y \cdot \frac{t}{a} + x \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{t}{a}}, x\right) \]

      10. lower-/.f64 80.5

          \[\leadsto \mathsf{fma}\left(y, \frac{t}{\color{blue}{a}}, x\right) \]

   4. Applied rewrites 80.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{t}{a}, x\right)} \]

   if 1.99999999999999989e183 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 85.1%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]

   3. Applied rewrites 97.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y \cdot \left(t - z\right)}{a}} \]
   5. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto y \cdot \color{blue}{\frac{t - z}{a}} \]

      2. sub-div N/A

         \[\leadsto y \cdot \left(\frac{t}{a} - \color{blue}{\frac{z}{a}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\frac{t}{a} - \frac{z}{a}\right) \cdot \color{blue}{y} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{t}{a} - \frac{z}{a}\right) \cdot \color{blue}{y} \]

      5. sub-div N/A

         \[\leadsto \frac{t - z}{a} \cdot y \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{t - z}{a} \cdot y \]

      7. lift--.f64 88.4

         \[\leadsto \frac{t - z}{a} \cdot y \]

   6. Applied rewrites 88.4%

      \[\leadsto \color{blue}{\frac{t - z}{a} \cdot y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 82.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - z}{a} \cdot y\\ t_2 := \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+183}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ (- t z) a) y)) (t_2 (/ (* y (- z t)) a)))
   (if (<= t_2 (- INFINITY)) t_1 (if (<= t_2 2e+183) (fma (/ y a) t x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((t - z) / a) * y;
	double t_2 = (y * (z - t)) / a;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 2e+183) {
		tmp = fma((y / a), t, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(t - z) / a) * y)
	t_2 = Float64(Float64(y * Float64(z - t)) / a)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 2e+183)
		tmp = fma(Float64(y / a), t, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 2e+183], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -inf.0 or 1.99999999999999989e183 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 83.6%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]

   3. Applied rewrites 98.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y \cdot \left(t - z\right)}{a}} \]
   5. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto y \cdot \color{blue}{\frac{t - z}{a}} \]

      2. sub-div N/A

         \[\leadsto y \cdot \left(\frac{t}{a} - \color{blue}{\frac{z}{a}}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\frac{t}{a} - \frac{z}{a}\right) \cdot \color{blue}{y} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{t}{a} - \frac{z}{a}\right) \cdot \color{blue}{y} \]

      5. sub-div N/A

         \[\leadsto \frac{t - z}{a} \cdot y \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{t - z}{a} \cdot y \]

      7. lift--.f64 91.6

         \[\leadsto \frac{t - z}{a} \cdot y \]

   6. Applied rewrites 91.6%

      \[\leadsto \color{blue}{\frac{t - z}{a} \cdot y} \]

   if -inf.0 < (/.f64 (*.f64 y (-.f64 z t)) a) < 1.99999999999999989e183

   1. Initial program 99.4%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]

   3. Applied rewrites 96.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)} \]

   4. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\frac{y}{a}, \color{blue}{t}, x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 77.4%

      \[\leadsto \mathsf{fma}\left(\frac{y}{a}, \color{blue}{t}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 76.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+231}:\\ \;\;\;\;\frac{-z}{a} \cdot y\\ \mathbf{elif}\;z \leq 2.55 \cdot 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot \frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.2e+231)
   (* (/ (- z) a) y)
   (if (<= z 2.55e+98) (fma (/ y a) t x) (* (- z) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.2e+231) {
		tmp = (-z / a) * y;
	} else if (z <= 2.55e+98) {
		tmp = fma((y / a), t, x);
	} else {
		tmp = -z * (y / a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.2e+231)
		tmp = Float64(Float64(Float64(-z) / a) * y);
	elseif (z <= 2.55e+98)
		tmp = fma(Float64(y / a), t, x);
	else
		tmp = Float64(Float64(-z) * Float64(y / a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.2e+231], N[(N[((-z) / a), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, 2.55e+98], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], N[((-z) * N[(y / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.19999999999999992e231

   1. Initial program 90.3%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot z}{a}} \]
   3. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto -1 \cdot \left(\frac{y}{a} \cdot \color{blue}{z}\right) \]

      2. associate-*l* N/A

         \[\leadsto \left(-1 \cdot \frac{y}{a}\right) \cdot \color{blue}{z} \]

      3. *-commutative N/A

         \[\leadsto z \cdot \color{blue}{\left(-1 \cdot \frac{y}{a}\right)} \]

      4. mul-1-neg N/A

         \[\leadsto z \cdot \left(\mathsf{neg}\left(\frac{y}{a}\right)\right) \]

      5. distribute-rgt-neg-out N/A

         \[\leadsto \mathsf{neg}\left(z \cdot \frac{y}{a}\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(\frac{z \cdot y}{a}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\frac{y \cdot z}{a}\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(y \cdot \frac{z}{a}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\frac{z}{a} \cdot y\right) \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \color{blue}{y} \]

      11. distribute-neg-frac2 N/A

          \[\leadsto \frac{z}{\mathsf{neg}\left(a\right)} \cdot y \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{z}{\mathsf{neg}\left(a\right)} \cdot \color{blue}{y} \]

      13. distribute-neg-frac2 N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot y \]

      14. distribute-neg-frac N/A

          \[\leadsto \frac{\mathsf{neg}\left(z\right)}{a} \cdot y \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(z\right)}{a} \cdot y \]

      16. lower-neg.f64 68.1

          \[\leadsto \frac{-z}{a} \cdot y \]

   4. Applied rewrites 68.1%

      \[\leadsto \color{blue}{\frac{-z}{a} \cdot y} \]

   if -2.19999999999999992e231 < z < 2.54999999999999994e98

   1. Initial program 94.6%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]

   3. Applied rewrites 97.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)} \]

   4. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\frac{y}{a}, \color{blue}{t}, x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 79.9%

      \[\leadsto \mathsf{fma}\left(\frac{y}{a}, \color{blue}{t}, x\right) \]

   if 2.54999999999999994e98 < z

   1. Initial program 89.4%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot \left(z - t\right)}{a}} \]

   4. Applied rewrites 75.0%

      \[\leadsto \color{blue}{\left(t - z\right) \cdot \frac{y}{a}} \]

   5. Taylor expanded in z around inf

      \[\leadsto \left(-1 \cdot z\right) \cdot \frac{\color{blue}{y}}{a} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(z\right)\right) \cdot \frac{y}{a} \]

      2. lower-neg.f64 64.9

         \[\leadsto \left(-z\right) \cdot \frac{y}{a} \]

   7. Applied rewrites 64.9%

      \[\leadsto \left(-z\right) \cdot \frac{\color{blue}{y}}{a} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 74.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+223}:\\ \;\;\;\;\frac{-z}{a} \cdot y\\ \mathbf{elif}\;z \leq 2.55 \cdot 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot \frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.9e+223)
   (* (/ (- z) a) y)
   (if (<= z 2.55e+98) (fma y (/ t a) x) (* (- z) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.9e+223) {
		tmp = (-z / a) * y;
	} else if (z <= 2.55e+98) {
		tmp = fma(y, (t / a), x);
	} else {
		tmp = -z * (y / a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.9e+223)
		tmp = Float64(Float64(Float64(-z) / a) * y);
	elseif (z <= 2.55e+98)
		tmp = fma(y, Float64(t / a), x);
	else
		tmp = Float64(Float64(-z) * Float64(y / a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.9e+223], N[(N[((-z) / a), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, 2.55e+98], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], N[((-z) * N[(y / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.9e223

   1. Initial program 90.5%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot z}{a}} \]
   3. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto -1 \cdot \left(\frac{y}{a} \cdot \color{blue}{z}\right) \]

      2. associate-*l* N/A

         \[\leadsto \left(-1 \cdot \frac{y}{a}\right) \cdot \color{blue}{z} \]

      3. *-commutative N/A

         \[\leadsto z \cdot \color{blue}{\left(-1 \cdot \frac{y}{a}\right)} \]

      4. mul-1-neg N/A

         \[\leadsto z \cdot \left(\mathsf{neg}\left(\frac{y}{a}\right)\right) \]

      5. distribute-rgt-neg-out N/A

         \[\leadsto \mathsf{neg}\left(z \cdot \frac{y}{a}\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(\frac{z \cdot y}{a}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\frac{y \cdot z}{a}\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(y \cdot \frac{z}{a}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\frac{z}{a} \cdot y\right) \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot \color{blue}{y} \]

      11. distribute-neg-frac2 N/A

          \[\leadsto \frac{z}{\mathsf{neg}\left(a\right)} \cdot y \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{z}{\mathsf{neg}\left(a\right)} \cdot \color{blue}{y} \]

      13. distribute-neg-frac2 N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{z}{a}\right)\right) \cdot y \]

      14. distribute-neg-frac N/A

          \[\leadsto \frac{\mathsf{neg}\left(z\right)}{a} \cdot y \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(z\right)}{a} \cdot y \]

      16. lower-neg.f64 66.9

          \[\leadsto \frac{-z}{a} \cdot y \]

   4. Applied rewrites 66.9%

      \[\leadsto \color{blue}{\frac{-z}{a} \cdot y} \]

   if -1.9e223 < z < 2.54999999999999994e98

   1. Initial program 94.7%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x - -1 \cdot \frac{t \cdot y}{a}} \]
   3. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x - \left(\mathsf{neg}\left(\frac{t \cdot y}{a}\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto x - \left(\mathsf{neg}\left(t \cdot \frac{y}{a}\right)\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto x - \left(\mathsf{neg}\left(t\right)\right) \cdot \color{blue}{\frac{y}{a}} \]

      4. fp-cancel-sign-sub-inv N/A

         \[\leadsto x + \color{blue}{t \cdot \frac{y}{a}} \]

      5. associate-/l* N/A

         \[\leadsto x + \frac{t \cdot y}{\color{blue}{a}} \]

      6. +-commutative N/A

         \[\leadsto \frac{t \cdot y}{a} + \color{blue}{x} \]

      7. *-commutative N/A

         \[\leadsto \frac{y \cdot t}{a} + x \]

      8. associate-/l* N/A

         \[\leadsto y \cdot \frac{t}{a} + x \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{t}{a}}, x\right) \]

      10. lower-/.f64 77.0

          \[\leadsto \mathsf{fma}\left(y, \frac{t}{\color{blue}{a}}, x\right) \]

   4. Applied rewrites 77.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{t}{a}, x\right)} \]

   if 2.54999999999999994e98 < z

   1. Initial program 89.4%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot \left(z - t\right)}{a}} \]

   4. Applied rewrites 75.0%

      \[\leadsto \color{blue}{\left(t - z\right) \cdot \frac{y}{a}} \]

   5. Taylor expanded in z around inf

      \[\leadsto \left(-1 \cdot z\right) \cdot \frac{\color{blue}{y}}{a} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(z\right)\right) \cdot \frac{y}{a} \]

      2. lower-neg.f64 64.9

         \[\leadsto \left(-z\right) \cdot \frac{y}{a} \]

   7. Applied rewrites 64.9%

      \[\leadsto \left(-z\right) \cdot \frac{\color{blue}{y}}{a} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 60.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+98}:\\ \;\;\;\;\left(-z\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;t\_1 \leq 500000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) a)))
   (if (<= t_1 -1e+98)
     (* (- z) (/ y a))
     (if (<= t_1 500000.0) x (* (/ y a) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (z - t)) / a;
	double tmp;
	if (t_1 <= -1e+98) {
		tmp = -z * (y / a);
	} else if (t_1 <= 500000.0) {
		tmp = x;
	} else {
		tmp = (y / a) * t;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (y * (z - t)) / a
    if (t_1 <= (-1d+98)) then
        tmp = -z * (y / a)
    else if (t_1 <= 500000.0d0) then
        tmp = x
    else
        tmp = (y / a) * t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (z - t)) / a;
	double tmp;
	if (t_1 <= -1e+98) {
		tmp = -z * (y / a);
	} else if (t_1 <= 500000.0) {
		tmp = x;
	} else {
		tmp = (y / a) * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y * (z - t)) / a
	tmp = 0
	if t_1 <= -1e+98:
		tmp = -z * (y / a)
	elif t_1 <= 500000.0:
		tmp = x
	else:
		tmp = (y / a) * t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y * Float64(z - t)) / a)
	tmp = 0.0
	if (t_1 <= -1e+98)
		tmp = Float64(Float64(-z) * Float64(y / a));
	elseif (t_1 <= 500000.0)
		tmp = x;
	else
		tmp = Float64(Float64(y / a) * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y * (z - t)) / a;
	tmp = 0.0;
	if (t_1 <= -1e+98)
		tmp = -z * (y / a);
	elseif (t_1 <= 500000.0)
		tmp = x;
	else
		tmp = (y / a) * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+98], N[((-z) * N[(y / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 500000.0], x, N[(N[(y / a), $MachinePrecision] * t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -9.99999999999999998e97

   1. Initial program 88.3%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot \left(z - t\right)}{a}} \]

   4. Applied rewrites 88.9%

      \[\leadsto \color{blue}{\left(t - z\right) \cdot \frac{y}{a}} \]

   5. Taylor expanded in z around inf

      \[\leadsto \left(-1 \cdot z\right) \cdot \frac{\color{blue}{y}}{a} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(z\right)\right) \cdot \frac{y}{a} \]

      2. lower-neg.f64 50.8

         \[\leadsto \left(-z\right) \cdot \frac{y}{a} \]

   7. Applied rewrites 50.8%

      \[\leadsto \left(-z\right) \cdot \frac{\color{blue}{y}}{a} \]

   if -9.99999999999999998e97 < (/.f64 (*.f64 y (-.f64 z t)) a) < 5e5

   1. Initial program 99.2%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 73.6%

      \[\leadsto \color{blue}{x} \]

   if 5e5 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 89.9%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{t \cdot y}{a}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{y \cdot t}{a} \]

      2. associate-/l* N/A

         \[\leadsto y \cdot \color{blue}{\frac{t}{a}} \]

      3. lower-*.f64 N/A

         \[\leadsto y \cdot \color{blue}{\frac{t}{a}} \]

      4. lower-/.f64 42.2

         \[\leadsto y \cdot \frac{t}{\color{blue}{a}} \]

   4. Applied rewrites 42.2%

      \[\leadsto \color{blue}{y \cdot \frac{t}{a}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto y \cdot \color{blue}{\frac{t}{a}} \]

      2. lift-/.f64 N/A

         \[\leadsto y \cdot \frac{t}{\color{blue}{a}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{y \cdot t}{\color{blue}{a}} \]

      4. associate-*l/ N/A

         \[\leadsto \frac{y}{a} \cdot \color{blue}{t} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{y}{a} \cdot \color{blue}{t} \]

      6. lift-/.f64 46.6

         \[\leadsto \frac{y}{a} \cdot t \]

   6. Applied rewrites 46.6%

      \[\leadsto \frac{y}{a} \cdot \color{blue}{t} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 59.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ t_2 := \frac{y}{a} \cdot t\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+98}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 500000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) a)) (t_2 (* (/ y a) t)))
   (if (<= t_1 -1e+98) t_2 (if (<= t_1 500000.0) x t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (z - t)) / a;
	double t_2 = (y / a) * t;
	double tmp;
	if (t_1 <= -1e+98) {
		tmp = t_2;
	} else if (t_1 <= 500000.0) {
		tmp = x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (y * (z - t)) / a
    t_2 = (y / a) * t
    if (t_1 <= (-1d+98)) then
        tmp = t_2
    else if (t_1 <= 500000.0d0) then
        tmp = x
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (z - t)) / a;
	double t_2 = (y / a) * t;
	double tmp;
	if (t_1 <= -1e+98) {
		tmp = t_2;
	} else if (t_1 <= 500000.0) {
		tmp = x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y * (z - t)) / a
	t_2 = (y / a) * t
	tmp = 0
	if t_1 <= -1e+98:
		tmp = t_2
	elif t_1 <= 500000.0:
		tmp = x
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y * Float64(z - t)) / a)
	t_2 = Float64(Float64(y / a) * t)
	tmp = 0.0
	if (t_1 <= -1e+98)
		tmp = t_2;
	elseif (t_1 <= 500000.0)
		tmp = x;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y * (z - t)) / a;
	t_2 = (y / a) * t;
	tmp = 0.0;
	if (t_1 <= -1e+98)
		tmp = t_2;
	elseif (t_1 <= 500000.0)
		tmp = x;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y / a), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+98], t$95$2, If[LessEqual[t$95$1, 500000.0], x, t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -9.99999999999999998e97 or 5e5 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 89.2%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{t \cdot y}{a}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{y \cdot t}{a} \]

      2. associate-/l* N/A

         \[\leadsto y \cdot \color{blue}{\frac{t}{a}} \]

      3. lower-*.f64 N/A

         \[\leadsto y \cdot \color{blue}{\frac{t}{a}} \]

      4. lower-/.f64 44.7

         \[\leadsto y \cdot \frac{t}{\color{blue}{a}} \]

   4. Applied rewrites 44.7%

      \[\leadsto \color{blue}{y \cdot \frac{t}{a}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto y \cdot \color{blue}{\frac{t}{a}} \]

      2. lift-/.f64 N/A

         \[\leadsto y \cdot \frac{t}{\color{blue}{a}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{y \cdot t}{\color{blue}{a}} \]

      4. associate-*l/ N/A

         \[\leadsto \frac{y}{a} \cdot \color{blue}{t} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{y}{a} \cdot \color{blue}{t} \]

      6. lift-/.f64 49.8

         \[\leadsto \frac{y}{a} \cdot t \]

   6. Applied rewrites 49.8%

      \[\leadsto \frac{y}{a} \cdot \color{blue}{t} \]

   if -9.99999999999999998e97 < (/.f64 (*.f64 y (-.f64 z t)) a) < 5e5

   1. Initial program 99.2%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 73.6%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 57.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ t_2 := y \cdot \frac{t}{a}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+98}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 500000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) a)) (t_2 (* y (/ t a))))
   (if (<= t_1 -1e+98) t_2 (if (<= t_1 500000.0) x t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (z - t)) / a;
	double t_2 = y * (t / a);
	double tmp;
	if (t_1 <= -1e+98) {
		tmp = t_2;
	} else if (t_1 <= 500000.0) {
		tmp = x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (y * (z - t)) / a
    t_2 = y * (t / a)
    if (t_1 <= (-1d+98)) then
        tmp = t_2
    else if (t_1 <= 500000.0d0) then
        tmp = x
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (z - t)) / a;
	double t_2 = y * (t / a);
	double tmp;
	if (t_1 <= -1e+98) {
		tmp = t_2;
	} else if (t_1 <= 500000.0) {
		tmp = x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y * (z - t)) / a
	t_2 = y * (t / a)
	tmp = 0
	if t_1 <= -1e+98:
		tmp = t_2
	elif t_1 <= 500000.0:
		tmp = x
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y * Float64(z - t)) / a)
	t_2 = Float64(y * Float64(t / a))
	tmp = 0.0
	if (t_1 <= -1e+98)
		tmp = t_2;
	elseif (t_1 <= 500000.0)
		tmp = x;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y * (z - t)) / a;
	t_2 = y * (t / a);
	tmp = 0.0;
	if (t_1 <= -1e+98)
		tmp = t_2;
	elseif (t_1 <= 500000.0)
		tmp = x;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(t / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+98], t$95$2, If[LessEqual[t$95$1, 500000.0], x, t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -9.99999999999999998e97 or 5e5 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 89.2%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{t \cdot y}{a}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{y \cdot t}{a} \]

      2. associate-/l* N/A

         \[\leadsto y \cdot \color{blue}{\frac{t}{a}} \]

      3. lower-*.f64 N/A

         \[\leadsto y \cdot \color{blue}{\frac{t}{a}} \]

      4. lower-/.f64 44.7

         \[\leadsto y \cdot \frac{t}{\color{blue}{a}} \]

   4. Applied rewrites 44.7%

      \[\leadsto \color{blue}{y \cdot \frac{t}{a}} \]

   if -9.99999999999999998e97 < (/.f64 (*.f64 y (-.f64 z t)) a) < 5e5

   1. Initial program 99.2%

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 73.6%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 39.2% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 x)

C

double code(double x, double y, double z, double t, double a) {
	return x;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x;
}

Python

def code(x, y, z, t, a):
	return x

Julia

function code(x, y, z, t, a)
	return x
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x;
end

Wolfram

code[x_, y_, z_, t_, a_] := x

Derivation

1. Initial program 93.5%

   \[x - \frac{y \cdot \left(z - t\right)}{a} \]

2. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x} \]
3. Step-by-step derivation

4. Applied rewrites 39.2%

   \[\leadsto \color{blue}{x} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025130 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F"
  :precision binary64
  (- x (/ (* y (- z t)) a)))